zbMATH — the first resource for mathematics

On \(\mathfrak F_n\)-normal subgroups of finite groups. (English. Russian original) Zbl 1230.20015
Sib. Math. J. 52, No. 2, 197-206 (2011); translation from Sib. Mat. Zh. 52, No. 2, 250-264 (2011).
All groups considered are finite. For a class \(\mathfrak F\) of groups, in the paper under review a subgroup \(H\) of a group \(G\) is defined to be \(\mathfrak F_n\)-normal in \(G\) if there exists a normal subgroup \(T\) of \(G\) such that \(HT\) is a normal subgroup of \(G\) and \((H\cap T)H_G/H_G\) is contained in the \(\mathfrak F\)-hypercenter \(Z_\infty^{\mathfrak F}(G/H_G)\) of \(G/H_G\), where \(H_G\) denotes the core of \(H\) in \(G\). We recall that the \(\mathfrak F\)-hypercenter is the largest normal subgroup of \(G\) whose \(G\)-chief factors are \(\mathfrak F\)-central (a chief factor \(H/K\) of \(G\) is \(\mathfrak F\)-central if \([H/K](G/C_G(H/K))\in\mathfrak F\)).
This concept is a generalization of \(c\)-normality (introduced by Y. Wang [in J. Pure Appl. Algebra 110, No. 3, 315-320 (1996; Zbl 0853.20015)]), \(\mathfrak F_n\)-supplementation (or \(\mathfrak F_c\)-normality) (considered by N. Yang and the first author [in Asian-Eur. J. Math. 1, No. 4, 619-629 (2008; Zbl 1176.20018)] and by A. Y. Alsheik Ahmad, J. J. Jaraden and A. N. Skiba [in Algebra Colloq. 14, No. 1, 25-36 ( 2007; Zbl 1126.20012)]) and \(\mathfrak F_h\)-normality (defined by X. Feng and the first author [in Front. Math. China 5, No. 4, 653-664 (2010; Zbl 1226.20011)]).
In this paper the authors use this new embedding property to obtain some criteria for supersolubility and \(p\)-nilpotency (\(p\) a prime) of groups, taking further some previous developments.

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D15 Finite nilpotent groups, \(p\)-groups
20D40 Products of subgroups of abstract finite groups
Full Text: DOI
[1] Guo W., The Theory of Classes of Groups, Science Press/Kluwer Academic Publishers, Beijing; New York; Dordrecht; Boston; London (2000).
[2] Robinson D. J. S., A Course in the Theory of Groups, Springer-Verlag, New York (1982). · Zbl 0483.20001
[3] Buckley J., ”Finite groups whose minimal subgroups are normal,” Math. Z., 15, 15–17 (1970). · Zbl 0202.02303 · doi:10.1007/BF01110184
[4] Srinivasan S., ”Two sufficient conditions for supersolubility of finite groups,” Israel J. Math, 35, No. 3, 210–214 (1980). · Zbl 0437.20012 · doi:10.1007/BF02761191
[5] Ramadan M., ”Influence of normality on maximal subgroups of Sylow subgroup of finite groups,” Acta Math. Hungar., 59, 107–110 (1992). · Zbl 0802.20019 · doi:10.1007/BF00052096
[6] Wang Y., ”C-normality of groups and its properties,” J. Algebra, 180, No. 3, 954–965 (1996). · Zbl 0847.20010 · doi:10.1006/jabr.1996.0103
[7] Yang N. and Guo W., ”On n-supplemented subgroups of finite groups,” Asian-European J. Math., 4, No. 1, 619–629 (2008). · Zbl 1176.20018 · doi:10.1142/S1793557108000485
[8] Shemetkov L. A. and Skiba A. N., Formations of Algebraic Systems [in Russian], Nauka, Moscow (1989). · Zbl 0667.08001
[9] Guo W., ”On -supplemented subgroups of finite groups,” Manuscripta Math., 127, 139–150 (2008). · Zbl 1172.20019 · doi:10.1007/s00229-008-0194-7
[10] Skiba A. N., ”On weakly s-permutable subgroups of finite groups,” J. Algebra, 315, No. 1, 192–209 (2007). · Zbl 1130.20019 · doi:10.1016/j.jalgebra.2007.04.025
[11] Shemetkov L. A., Formations of Finite Groups [in Russian], Nauka, Moscow (1978). · Zbl 0496.20014
[12] Guo W., Wang Y., and Shi L., ”Nearly S-normal subgroups of a finite group,” J. Algebra Discrete Struct., 6, 95–106 (2008). · Zbl 1170.20015
[13] Huppert B., Endliche Gruppen. I, Springer-Verlag, Berlin, Heidelberg, and New York (1967).
[14] Gorenstein D., Finite Groups, Chelsea Publ. Company, New York (1980). · Zbl 0463.20012
[15] Li D. and Guo X., ”The influence of c-normality of subgroups on the structure of finite groups. II,” Comm. Algebra, 26, No. 6, 1913–1922 (1998). · Zbl 0907.20052 · doi:10.1080/00927879808826342
[16] Ahmad Alsheik A., ”Finite groups with given c-permutable subgroups,” Algebra Discrete Math., 2, 74–85 (2004). · Zbl 1067.20018
[17] Guo W., Shum K. P., and Skiba A. N., ”G-covering subgroup systems for the classes of supersoluble and nilpotent groups,” Israel J. Math, 138, 125–138 (2003). · Zbl 1050.20009 · doi:10.1007/BF02783422
[18] Ramdan M., ”Influence of normality on maximal subgroups of Sylow subgroup of finite groups,” Acta Math. Hungar., 73, 335–342 (1996). · Zbl 0930.20021 · doi:10.1007/BF00052909
[19] Wei H., ”On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups,” Comm. Algebra, 29, No. 5, 2193–2200 (2001). · Zbl 0990.20012 · doi:10.1081/AGB-100002178
[20] Guo X. and Shum K. P., ”On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups,” Arch. Math., 80, No. 6, 561–569 (2003). · Zbl 1050.20010 · doi:10.1007/s00013-003-0810-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.